Rates of clustering in Strassen's LIL for Brownian motion
type: Item
instance of: scholarly article
title: Rates of clustering in Strassen's LIL for Brownian motion
zbMATH Open document ID: 0724.60034
DOI: 10.1007/BF01258738
author: Victor W. Goodman
author: Jim Kuelbs
published in: Journal of Theoretical Probability
publication date: 1991-01-01T00:00:00Z
review text: For standard Brownian motion \(\{W(t): 0\leq t<\infty \}\) let \(\eta_ n(t)=W(nt)(2nL_ 2n)^{-1/2}\), where \(t\in [0,1]\), \(n\in {\mathbb{N}}\) and \(L_ mn=L(L_{m-1}n)\), \(Lx=\max \{1,\log_ ex\}\). For a set \(A\subset C[0,1]\) denote \(A^{\epsilon}=\{g\in C[0,1]:\;\inf_{f\in A}\| g- f\| <\epsilon \},\) where \(\epsilon >0\) and \(\| \cdot \|\) is the uniform norm. The following generalizations of \textit{K. Grill}'s result [Probab. Theory Relat. Fields 74, 583-589 (1987; Zbl 0592.60022)] are obtained. If \({\mathcal K}\) is the Strassen's set, \(\epsilon_ n=\gamma (L_ 3n/L_ 2n)^{2/3}\), then for \(\gamma >0\) sufficiently large, \(P(\eta_ n\in {\mathcal K}^{\epsilon_ n}\) eventually)=1. If \(\epsilon_ n=\gamma (L_ 2n)^{-2/3}\), then for \(\gamma >0\) sufficiently small \(P(\eta_ n\not\in {\mathcal K}^{\epsilon_ n}\quad i.o.)=1.\) Introduce the random sets \(E_{\xi (n),n}=\{W(k(\cdot))(2kL_ 2n)^{-1/2}:\;\xi (n)\leq k\leq n\}.\) If \(\xi (n)\leq n^{1/4}\) and \(\epsilon_ n=\gamma (L_ 2n)^{-2/3}\), then for \(\gamma >0\) sufficiently large, \(P({\mathcal K}\subseteq (E_{\xi (n),n})^{\epsilon_ n}\) eventually)=1 and for \(\gamma >0\) sufficiently small \(P({\mathcal K}\subseteq (E_{1,n})^{\epsilon_ n}\) eventually)=0. Assume \(L_ 2\xi (n)\sim L_ 2n\) as \(n\to \infty\). Then for \(\epsilon_ n=\gamma (L_ 3n/L_ 2n)^{2/3}\) with \(\gamma >0\) sufficiently large, \(P(E_{\xi (n),n}\subseteq {\mathcal K}^{\epsilon_ n}\) eventually)=1.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 4194838
zbMATH Keywords: Brownian motion
zbMATH Keywords: rates of clustering
reviewed by: Alexander V. Bulinski
MaRDI profile type: Publication
cites work: On the speed of convergence in Strassen's law of the iterated logarithm
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full work available at URL: https://doi.org/10.1007/bf01258738
OpenAlex ID: W2023588110
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description: scientific article; zbMATH DE number 4194838